For example, a general quadratic function is commonly written as:
which makes the function-argument status of x (and by extension the constancy of a, b and c) clear. In this example a, b and c are coefficients of the polynomial. Since c occurs in a term that does not involve x, it is called the constant term of the polynomial and can be thought of as the coefficient of x0. More generally, any polynomial term or expression of degree zero is a constant.:18
A constant may be used to define a constant function that ignores its arguments and always gives the same value. A constant function of a single variable, such as , has a graph of a horizontal straight line parallel to the x-axis. Such a function always takes the same value (in this case, 5), because its argument does not appear in the expression defining the function.
The context-dependent nature of the concept of "constant" can be seen in this example from elementary calculus:
"Constant" means not depending on some variable; not changing as that variable changes. In the first case above, it means not depending on h; in the second, it means not depending on x. A constant in a narrower context could be regarded as a variable in a broader context.
Some values occur frequently in mathematics and are conventionally denoted by a specific symbol. These standard symbols and their values are called mathematical constants. Examples include:
In calculus, constants are treated in several different ways depending on the operation. For example, the derivative of a constant function is zero. This is because the derivative measures the rate of change of a function with respect to a variable, and since constants, by definition, do not change, their derivative is hence zero.
Conversely, when integrating a constant function, the constant is multiplied by the variable of integration. During the evaluation of a limit, the constant remains the same as it was before and after evaluation.
Integration of a function of one variable often involves a constant of integration. This arises due to the fact that the integral operator is the inverse of the differential operator, meaning that the aim of integration is to recover the original function before differentiation. The differential of a constant function is zero, as noted above, and the differential operator is a linear operator, so functions that only differ by a constant term have the same derivative. To acknowledge this, a constant of integration is added to an indefinite integral; this ensures that all possible solutions are included. The constant of integration is generally written as 'c', and represents a constant with a fixed but undefined value.
If f is the constant function such that for every x then